Normed Division Algebra
Real algebra whose norm multiplies and in which every nonzero element is invertible β only four exist: β, β, β, and π.
Specifications & Metrics
Core Principles & Analysis
- A finite-dimensional real algebra with a nondegenerate positive-definite quadratic form that respects multiplication.
- Hurwitz's theorem, published posthumously in 1923, solves the classification problem for these algebras.
- The only examples occur in dimensions 1, 2, 4, and 8: the real numbers, complex numbers, quaternions, and octonions.
Sources